Given the information, we can construct a polynomial $P(x)$ of degree 5 based on the roots and their multiplicities.

1. Root at $x = 3$ with multiplicity 2: This gives the factor $(x - 3)^2$.
2. Root at $x = 0$ with multiplicity 2: This gives the factor $x^2$.
3. Root at $x = -3$ with multiplicity 1: This gives the factor $(x + 3)$.

Since the polynomial has a leading coefficient of 1, we can multiply these factors directly.

Constructing $P(x)$:

$P(x) = (x - 3)^2 \cdot x^2 \cdot (x + 3)$

Expanding this expression is optional but often helpful. The polynomial is:

$P(x) = x^2 (x - 3)^2 (x + 3)$

This is a valid formula for $P(x)$ that satisfies all given conditions.

Would you like me to expand this further, or do you have questions on the steps?

Here are related questions that might deepen your understanding:

1. How do you find the degree of a polynomial given its roots and multiplicities?
2. How does the leading coefficient affect the end behavior of a polynomial?
3. What are other properties of a polynomial determined by its roots and multiplicities?
4. How would the formula change if one of the roots had a higher multiplicity?
5. How do we verify the correctness of this polynomial?

Tip: When constructing polynomials, always check that the sum of the multiplicities matches the desired degree.